On the global attractivity and the periodic character of some difference equations. (English) Zbl 0993.39008

El-Metwally, H.; Grove, E. A.; Ladas, G.; Voulov, H. D.

J. Difference Equ. Appl. 7, No. 6, 837-850 (2001).

Asymptotic properties of solutions of the \(k\)-th order difference equation \[ x_{n+1}=\frac{A_0}{x_n}+\frac{A_1}{x_{n-1}}+\dots+ \frac{A_{k-1}}{x_{n-k+1}},\quad n\in \mathbb N=\{0,1,\dots\} \tag{*} \] are investigated. It is shown that under some restrictions on the numbers \(A_0,\dots,A_{k-1}\) every positive solution of (*) converges to a \(p\)-periodic solution, where the period \(p\) is determined in terms of the coefficients \(A_0,\dots,A_{k-1}\). The main result of the paper reads as follows.
Theorem. Let \(A_0,\dots,A_{k-1}\) be nonnegative real numbers and suppose that the set \(J=\{j\geq 1:\;A_{j-1}>0\}\) is nonempty. Set \(L=\{i+j: i,j\in J\}\), and let \(p=2(\langle L\rangle +1)-\langle L\rangle/\langle J\rangle\), where \(\langle \cdot\rangle\) denotes the greatest common divisor of the elements of the set indicated. Then every positive solution of (*) converges to a periodic solution of (*) with (not necessarily prime) period \(p\). Moreover, there exist solutions of (*) which are periodic with prime period \(p\).

Reviewer: Ondřej Došlý (Brno)

MSC:

39A12	Discrete version of topics in analysis
37C70	Attractors and repellers of smooth dynamical systems and their topological structure
39A11	Stability of difference equations (MSC2000)
39A10	Additive difference equations

Keywords:

global convergence; periodic solutions; rational recursive sequences; global attractivity; difference equation; positive solution